scholarly journals Transition curves in a parametrically excited pendulum with a force of elliptic type

2012 ◽  
Vol 468 (2148) ◽  
pp. 3995-4007 ◽  
Author(s):  
Si Mohamed Sah ◽  
Brian Mann
Keyword(s):  
Numerical Integration ◽  
Elliptic Function ◽  
Harmonic Balance ◽  
Floquet Theory ◽  
Parameter Plane ◽  
Elliptic Type ◽  
Mathieu’S Equation ◽  
Balance Analysis ◽  
The Stability ◽  
Transition Curves

This article investigates the equilibria and stability of a pendulum when the support has a prescribed motion defined by an elliptic function. Stability charts are generated in the parameter plane for different values of the elliptic function modulus. Numerical integration and Floquet theory are used to generate stability charts that are later obtained through harmonic balance analysis. It is shown that the size and location of the instability tongues is directly linked to the elliptic function modulus. Comparisons are also made between the stability charts of Mathieu's equation and those of the pendulum when the prescribed motion is defined by an elliptic function.

scholarly journals Calculating periodic vibrations of nonlinear autonomous multi-degree-of-freedom systems by the incremental harmonic balance method

2004 ◽  
Vol 26 (3) ◽  
pp. 157-166
Author(s):  
Nguyen Van Khang ◽  
Thai Manh Cau
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Van Der Pol ◽  
Incremental Harmonic Balance ◽  
The Stability

In this paper the incremental harmonic balance method is used to calculate periodic vibrations of nonlinear autonomous multip-degree-of-freedom systems. According to Floquet theory, the stability of a periodic solution is checked by evaluating the eigenvalues of the monodromy matrix. Using the programme MAPLE, the authors have studied the periodic vibrations of the system multi-degree van der Pol form.

Stability Boundary for Pseudo-Random Parametric Excitation of a Linear Oscillator

1983 ◽  
Vol 105 (3) ◽  
pp. 326-331 ◽  
Author(s):  
D. Watt ◽  
A. D. S. Barr
Keyword(s):  
Numerical Integration ◽  
Monodromy Matrix ◽  
Stability Boundary ◽  
Gaussian White Noise ◽  
Stability Diagram ◽  
Phase Variable ◽  
Null Solution ◽  
Frequency Spacing ◽  
Mathieu’S Equation ◽  
The Stability

Stability bounds are outlined for the null solution of the equation describing the response of a linear damped oscillator excited through periodic coefficients, the excitation being a form of Rice noise comprising equal amplitude sinusoids with frequencies at equal intervals in the vicinity of twice the natural frequency of the system, but with pseudo-random initial phases. Stability was investigated by the monodromy matrix method, which is exact apart from errors due to numerical integration, and by the approximate method due to R. A. Struble, which replaces the dependent variable by its amplitude and a phase variable. Struble’s method gives the main features of the stability diagram and leads to faster and more robust numerical integration with potential advantages for nonlinear and several degree-of-freedom systems, but loses much of the detail. When the frequency spacing is relatively large, the stability diagram is closely related to that for Mathieu’s equation, but the detailed shape becomes very complicated as the frequency spacing decreases. Quantitative comparison with the corresponding boundary for Gaussian white noise excitation shows very approximate equivalence.

A Floquet-Based Analysis of Parametric Excitation Through the Damping Coefficient

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Fatemeh Afzali ◽  
Gizem D. Acar ◽  
Brian F. Feeny
Keyword(s):  
Floquet Theory ◽  
Initial Conditions ◽  
Harmonic Balance Method ◽  
Damping Coefficient ◽  
Periodic Part ◽  
Response Characteristics ◽  
Floquet Solution ◽  
Mathieu’S Equation ◽  
Truncated Fourier Series ◽  
The Stability

Abstract The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu’s equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.

Asymmetric Mathieu equations

2006 ◽  
Vol 462 (2070) ◽  
pp. 1643-1659 ◽  
Author(s):  
Amol Marathe ◽  
Anindya Chatterjee
Keyword(s):  
Periodic Solutions ◽  
Mathieu Equation ◽  
Parameter Plane ◽  
Pendulum System ◽  
Strongly Nonlinear ◽  
Numerical Studies ◽  
The Stability ◽  
Elastic Restraints ◽  
Vertical Base ◽  
Transition Curves

An inverted pendulum with asymmetric elastic restraints (e.g. a one-sided spring), when subjected to harmonic vertical base excitation, on linearizing trigonometric terms, is governed by an asymmetric Mathieu equation. This system is parametrically forced and strongly nonlinear (linearization for small motions is not possible). However, solutions are scaleable: if x ( t ) is a solution, then so is αx ( t ) for any real α >0. We numerically study the stability regions in the parameter plane of this system for a fixed degree of asymmetry in the elastic restraints. A Lyapunov-like exponent is defined and numerically evaluated to find these regions of stable and unstable behaviour. These numerics indicate that there are infinitely many possibilities of instabilities in this system that are missing in the usual or symmetric Mathieu equation. We find numerically that there are periodic solutions at the boundaries of stable regions in the parameter plane, analogous to the symmetric Mathieu equation. We compute and plot several of these solution branches, which provide a relatively simpler means of computing the stability transition curves of this system. We prove theoretically that such periodic solutions must exist on all stability boundaries. Our theoretical results apply to the asymmetric Hill's equation, of which the pendulum system is a special case. We demonstrate this with numerical studies of a more general asymmetric Mathieu equation.

Harmonic Balance Analysis of General Squeeze Film Damped Multidegree-of-Freedom Rotor Bearing Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 499-507 ◽  
Author(s):  
E. J. Hahn ◽  
P. Y. P. Chen
Keyword(s):  
Harmonic Balance ◽  
Reference Frames ◽  
Squeeze Film ◽  
Equilibrium Solutions ◽  
Rotor Bearing ◽  
Balance Analysis ◽  
Force Components ◽  
Lift Off ◽  
Rotating Reference Frames ◽  
The Stability

Squeeze film dampers introduce nonlinear motion dependent damper forces into otherwise linear rotor bearing systems, thereby considerably complicating their analysis. Noncircular orbit type dampers, such as unsupported or uncentralized dampers, have generally necessitated transient solutions, which are computationally prohibitive for design studies of large order systems, particularly for systems with low damping. By utilizing harmonic balance with appropriate condensation, it is possible to considerably reduce the number of simultaneous nonlinear equations inherent to this approach. The stability (linear) of the equilibrium solutions may be conveniently evaluated using Floquet theory, particularly if the damper force components are evaluated in fixed, rather than rotating, reference frames. The versatility of this technique is illustrated on systems of increasing complexity with and without damper centralizing springs. Of particular interest, is its applicability to unsupported systems illustrating how such systems can lift off and, with further increase in speed, the damper forces can be linearized about the orbit center.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

A TWELFTH-ORDER FOUR-STEP FORMULA FOR THE NUMERICAL INTEGRATION OF THE ONE-DIMENSIONAL SCHRÖDINGER EQUATION

2003 ◽  
Vol 14 (08) ◽  
pp. 1087-1105 ◽  
Author(s):  
ZHONGCHENG WANG ◽  
YONGMING DAI
Keyword(s):  
Schrödinger Equation ◽  
Numerical Integration ◽  
Schrodinger Equation ◽  
Numerical Test ◽  
New Method ◽  
Step Method ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Order Four

A new twelfth-order four-step formula containing fourth derivatives for the numerical integration of the one-dimensional Schrödinger equation has been developed. It was found that by adding multi-derivative terms, the stability of a linear multi-step method can be improved and the interval of periodicity of this new method is larger than that of the Numerov's method. The numerical test shows that the new method is superior to the previous lower orders in both accuracy and efficiency and it is specially applied to the problem when an increasing accuracy is requested.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

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